Digamma Function of One Half

Theorem

$\map \psi {\dfrac 1 2} = -\gamma - 2 \ln 2$

where:

$\gamma$ denotes the Euler-Mascheroni constant.

Proof 1

$\ds \map \psi {\frac 1 2}$

$=$

$\ds -\gamma - \ln 4 - \frac \pi 2 \map \cot {\frac 1 2 \pi} + 2 \sum_{n \mathop = 1}^0 \map \cos {\frac {2 \pi n} 2} \map \ln {\map \sin {\frac {\pi n} 2} }$

Gauss's Digamma Theorem

$\ds $

$=$

$\ds -\gamma - \ln 4$

Cotangent of Right Angle, noting also that the summation is an empty sum

$\ds $

$=$

$\ds -\gamma - 2 \ln 2$

Logarithm of Power

$\blacksquare$

Proof 2

$\ds \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}$

$=$

$\ds -\paren {n - 1} \gamma - n \ln n$

Digamma Additive Formula: Corollary

$\ds \leadsto \ \ $

$\ds \sum_{k \mathop = 1}^{2 - 1} \map \psi {\frac k 2}$

$=$

$\ds -\paren {2 - 1} \gamma - 2 \ln 2$